Walter Schachermayer

Weighted norm inequalities and hedging in incomplete markets.

F. Delbaen, P. Monat, W. Schachermayer, M. Schweizer, C. Stricker
Finance and Stochastics, Vol. 1 (1997), No. 3, pp. 181-227. 

Abstract:

Let $X$ be an $\bf R ^d$-valued special semimartingale on a probability space $(\O, \CF , (\CF _t)_{0 \leq t \leq T} , P)$ with canonical decomposition $X=X_0+M+A$ and $G_T(\T)$ the space of all random variables $(\t \IS X)_T$ where $\t$ is a predictable $X$-integrable process such that $\t \IS X$ is in the space $\CS ^2$ of semimartingales. We shall investigate under which conditions (on the semimartingale $X$) the space $G_T(\T)$ is closed in $\CL ^2(\O, \CF ,P)$. The question, wether $G_T(\T)$ is closed, arises naturally in the applications to Mathematical Finance. We give necessary and/or sufficient conditions for the closedness of $G_T(\T)$ in $\CL ^2(P)$. Most of these conditions deal with $BMO$-martingales and reverse H\"older inequalities which are equivalent to weighted norm inequalities. By means of these last inequalities, we also extend previous results on the F\"ollmer-Schweizer decomposition.

Keywords:

Semimartingales, Stochastic Integrals, Reverse H\"older Inequalities, $BMO$ Space, Weighted Norm Inequalities, F\"ollmer-Schweizer Decomposition

Preprints:

Publications marked with have appeared in refereed journals.

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